hepl summation

[late meteor]

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[magic robin]

Do without sandiwch theorem, conversion of summation to integration

[burnt sapphire]

So do it without both, or do it with riemann sums?

[magic robin]

do it withouit both

[burnt sapphire]

💀

You know the finite sum won't be a nice result, right

You're dealing with harmonics

[magic robin]

was dat

oh

oh

i understand

i mean

[magic robin]

is there some way i can break this series in 2 harmonics??

i can calculate that

[burnt sapphire]

Nope lol

Not 2 harmonics, you can notice that you have $\frac{r}{r+c}=1-\frac{c}{r+c}$

[little marten]

Kocher

[magic robin]

yes, still a single harmonic

so

there isnt another way

except those 2?

[burnt sapphire]

Nope

Unless you want to bash your head in

[magic robin]

nah im fine 💀

can you

teach me

reimann sum basic

[burnt sapphire]

Sure?

I'll give you an easy one

$\lim_{n\to\infty}\sum_{k=1}^n\frac{k}{n^2+k^2}$

[little marten]

Kocher

[magic robin]

what are the prerequisites??

i know limits

[burnt sapphire]

Just understanding the concept of the sum, I guess

[magic robin]

i know nothing of reimann sums though,

[burnt sapphire]

$\int_a^bf(x)dx=\lim_{n\to\infty}\sum_{k=1}^n\frac{b-a}{n}f\qty(a+\frac{b-a}n\cdot k)$

[little marten]

Kocher

[magic robin]

i just know theres something like r/n

[magic robin]

this seems familiar

[burnt sapphire]

Obviously I memorized it but I feel that there is no need to, so long as you get the infinite rectangle strip concept

[magic robin]

wait

[magic robin]

yes i understood

divided a-b into n (infinite here) rectangles and multiplied it by its height ( the value of function ) at every point

[magic robin]

shall i try to apply it in my question now?

[magic robin]

is the answer 1/4 ?

[burnt sapphire]

No

Show your work

[burnt sapphire]

Yeah mb

I'm studying

[magic robin]

im confused how to turn this summatoin into the

general form yu showed

[burnt sapphire]

||divide top and bottom by k^2||

[magic robin]

1/k becomes dx?

[burnt sapphire]

Yes

[magic robin]

and because summation starts from 1

fx = 1/x^2 ??

no no no wait

[magic robin]

im confuised 😭

im unable to compare it to the general form

[burnt sapphire]

Oh I screwed up lol

$\lim_{n\to\infty}\sum_{k=1}^n\frac{n}{n^2+k^2}$

[little marten]

Kocher

[burnt sapphire]

Or ok, that works still

^

[magic robin]

the previous one is solvable too

dividing by n^2 in numerator and enominator

it becomes xdx/1+x^2

0 to 1

[burnt sapphire]

Yep

[magic robin]

YAYY

thankyou sm

+close

[spark crypt]

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[burnt sapphire]

You're welcome

[spark crypt]

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